Science 7 - LAND and SEA BREEZE and its Characteristics
Math Discourse colloquium with Dr. Lucianna de Oliveira and Ms. Judith O'Loughlin!
1. Summing up Math Language:
Frameworks, Activities and Ideas to
Empower
Brenda Custodio– ESL Consultant, Ohio State University, custodio.1@osu.edu
Luciana de Oliveira– Associate Professor, University of Miami, ludeoliveira@miami.edu
Judith O'Loughlin– Consultant, Language Matters, LLC, joeslteach@aol.com
Kate Reynolds– Title III Grant Curriculum Designer, Missouri State University, katerey523@gmail.com
TESOL International Association, Seattle, WA, March 22, 2017.
2. Session Objectives
Participants learn to
1. analyze key mathematic vocabulary, functions and discourse patterns common at
different grade levels,
2. engage ELLs with vocabulary, phrase and sentence construction (i.e., sentence
level to discourse level competencies),
3. acquire a framework to transition from key words to word problems,
4. debate the “universality” of math (e.g., worldwide calendars, different date systems,
math symbols/notation, long division, metric system, military time, etc.), and
5. develop literacy strategies for helping ELs develop better math skills (e.g., math
notebooks, think alouds).
3. Math Language:
Discourse Patterns,
Linguistic
Functions and
Vocabulary.
›A Professor and teacher educator in TESL/TEFL,
currently works with in-service educators at Missouri
State University.
›Publications:
–Reynolds, K.M. (March, 2015). Approaches to inclusive
English classrooms: A teacher’s handbook for content
based instruction. Bristol: Multilingual Matters.
–Reynolds, K.M. (Ed.) (in press). Vocabulary volume of the
ELT Encyclopedia. Alexandria, VA: TESOL and Wiley.
–Reynolds, K.M. (in press). Developing content-based
objectives from academic standards. Upcoming chapter
in M.A. Snow and D. Brinton (Eds.), Handbook of Content-
Based ESL Instruction, 2nd edition. Cengage.
–Reynolds, K. M. and Jiao, J.J. (2012). Aha! Measuring
Teachers’ Content-Based Learning. MinneWITESOL
Journal, (29): 127-153.
–Reynolds, K. M. (2010). Exploration: One journey of
integrating content and language objectives. In J.
Nordmeyer & S Barduhn (Eds.), Integrating Language and
Content. Alexandria, VA: TESOL.
›
Dr. Kate Mastruserio
Reynolds– Title III Grant
Curriculum Designer,
Missouri State University,
katerey523@gmail.com
4. Objective
›Participants will be able to analyze key
mathematic vocabulary, functions and
discourse patterns common at different
grade clusters.
6. 2 + 3 = 5
What are some ways
that the teacher, text,
or test might phrase
this equation?
Possible answers:
•How many altogether?
•How many in all?
•How much is 3 and 2?
•What is the sum of…?
•What is 2 plus 3?
•Add the two numbers.
•Three squares and two more
are…
•Three plus two equals…
7. Common Types of Mathematics Vocabulary
Slavit, D., & Ernst-Slavit, G. (2007).
17. A Whole Lot of Words!
Tiers of Vocabulary
http://www.learningunlimitedllc.com/wp-content/uploads/2013/05/tiered-3.png
18. Tiers of Vocabulary
›Tier 1: Common, everyday words. Most children know when they enter school in
their native language(s). Bilingual children would know these words in one or both
languages.
›Examples: dog, ball, mom, dad, house, car
Beck, McKeown, and Kucan (2013)
https://images-na.ssl-images-amazon.com/images/G/01/img15/pet-products/small-
tiles/23695_pets_vertical_store_dogs_small_tile_8._CB312176604_.jpg
19. Tiers of Vocabulary
›Tier 2: Words that are employed in many content areas. Important for students
to know and understand long-term. Essential for academics.
›Academic and cognitive processing words. Student will see repetitively in texts,
texts, and oral discourse throughout schooling.
›Examples: describe, justify, contrast, elaborate
Beck, McKeown, and Kucan (2013)
http://p2cdn4static.sharpschool.com/UserFiles/Servers/Server_87286/Image/Vridder/Staff/BloomRevised
Taxonomy.jpg
20. Tiers of Vocabulary
›Tier 3: Content-specific vocabulary. The bolded, defined words in textbooks and
glossaries. Terminology and jargon.
›Words for learning a specific academic topic. Necessary for learning the academic
concepts and building students' background knowledge.
Beck, McKeown, and Kucan (2013) http://static.wixstatic.com/media/5cd6ef_b39fcd12c5a34ea7ae8ca26e54233076.jpg_srz_435_28
3_85_22_0.50_1.20_0.00_jpg_srz
21. Task: Analysis of Math Vocabulary
›Directions: Using the Math text/pages provided, please identify and categorize the academic
vocabulary into three sections using the 3 Tiers that would be new encounters for ELs.
Tier 1 Tier 2 Tier 3
22. A Whole Lot of Words!
Bernier’s Model of Academic Vocabulary
›Content terms—routinely occur in lectures and textbooks as “common
knowledge” references to course material within the disciple.
–“regular” math terms and jargon,
–archaic language (e.g., abacus, trope, addend),
–non-history terms or terms borrowed from other fields (e.g., lemma, root,
kite, harmonic mean),
–obscure acronyms and symbols (e.g., AAS, e, GCF, GLB, IQR (see more at
http://www.mathwords.com/a_to_z.htm), and
–non-English vocabulary (e.g., alpha, chi, gamma, eta, pi, epsilon, quotient).
Bernier, 1997, pp. 96-97.
23. Bernier’s Model of Academic Vocabulary
›Language terms—refers to vocabulary technically outside the boundaries
of the content, but that frequently finds its way into course lectures, readings,
and assignments.
–metaphors (offering the olive branch),
–colloquial usages (fell on deaf ears),
–class-based constructions (syllabus, office hours, mortgage, stocks and
bonds), and
–cultural idioms (two for one, ballpark figure, my 2 cents, a dime a dozen, a
penny saved is a penny earned, “classic” works of fiction allusions).
Bernier, 1997, pp. 96-97.
24. Bernier’s Model of Academic Vocabulary
›Language Masking Content—includes the fluid boundaries between the two
previous categories.
›Terms appropriated by teachers and scholars that hide content due to
–variant or multiple meanings (e.g., area, continuous, functions, place value,
right),
–unfamiliar metaphors (8-hour day/24-hour day; > sign is like a hungry
alligator; numerator is like the top cookie in an Oreo) and
–oxymorons (constant variable, bigger half, random order, sharp curve). (See
more at http://www.english-for-students.com/Mathematics.html).
Bernier, 1997, pp. 96-97.
25. Task: Analysis of Math Vocabulary
›Directions: Using the Math text/pages provided, please identify and categorize the academic
vocabulary into three sections using Bernier’s categories.
Content Terms Language Terms Content Masking Language
26. How should we teach the words?
Evidence-Based Teaching of Academic Vocabulary
What does current research tell us about teaching vocabulary?
1. Use frequency lists– Nation & Waring, 1997
2. Teach vocabulary explicitly –Marzano & Pickering, 2005
3. Actively engage students –Folse, 2004
4. Use repetition and multiple exposures through practice and materials –
McKeown et al., 1985
5. Use rich oral language- Carlo, August & Snow, 2005
6. Employ narrow, extensive reading for incidental learning- Day, Omura,
Hiramatsu, 1991; Schmitt, & Carter, 2000
28. 6 Steps for Teaching Academic Vocabulary
1. Provide a description, explanation, or
example of the new term.
2. Ask students to restate the description,
explanation, or example in their own words.
3. Ask students to construct a picture,
pictograph, or symbolic representation of the
term.
4. Ask students to contextualize the term.
Where might they see it in their lives?
5. Practice the term often in a short space of
time by
1. Vocabulary notebooks
2. Graphic organizers (Frayer’s Model)
3. Think-pair-share about terms
4. Play games with the terms
Adapted from Marzano, R. J. (2004).
http://3.bp.blogspot.com/-
VbZV_5I0J2A/UAMIA26DCdI/AAAAAAAAPKw/ADNDLNyIeHU/s1543/IMG_3
120.JPG
29. Pre-Teaching Math Vocabulary?
A Unique Feature of Math Vocabulary
›The academic vocabulary of math is
different from other content areas,
because the math vocabulary and
definitions are intertwined.
›It is difficult to define some math terms
through pre-teaching.
›The reason for this is that math terms
describe a concept/activity.
›One cannot teach the term without
teaching the mathematic concept.
30.
31. Task: Defining Math Vocabulary
›Directions:
1. Choose one term from the text book.
2. Write the textbook’s definition of the term
3. Write a formal definition of the term
Term + class/category + differentiation from other items in
category (Spaghetti is a kind of pasta that is long and thin and
usually served with sauce.)
4. Write an informal, or learner friendly, definition of the term.
5. Choose a visual or set of visuals you would use to help
explain this term. Draw it/them.
32. Patterns of Text: Discourse Patterns
›Directions:
›Think-Pair-Share
›Questions:
›What are the patterns of language at
the sentence level?
›What are the types of sentences
present in math texts?
›What have you noticed about the
language of math texts?
At the Sentence Level
33. Grammar in Math Sentences
›Incomplete sentences/phrases
–A number minus 6
›Simple sentences
–Maddy has two cupcakes.
›If/then constructions
–If Tariq has 5 and Juan has 3, then
what do they have all together?
›Wh- questions & Inverted subject
–How many are there?
–Does she have 10 or 12?
›Compounds and Series
–6 is 2 greater than 4 and five times as
high as
›Reversals
–The number a is five less than b.
Correct equation: a = b -5
34. What other grammatical patterns occur in math discourse?
› There is/are- dummy subject
› Present tense; timeless present
› Prepositions are key to
comprehension.
› Abbreviations, symbols, labels
› 2nd and 3rd person
Chamot, 2009; Li, 2012
http://resultncutoff.in/wp-content/uploads/2015/03/maths-ftr.jpg
35. Task: Identifying Patterns at the Sentence Level
›Directions: Using your math
book/pages,
–identify 3-4 patterns at the
sentence level, and
–2 sentence starters.
http://mathtrailblazers.uic.edu/mtb4-curriculum/components/student-materials/
36. Simplifying Fractions
There are two major difficulties when simplifying fractions. One is finding a number that is a common factor when it's
not "obvious", and the other is simplifying completely. To deal with either style problem it is helpful to have a
plan and approach each problem systematically.
The easiest way is to try dividing both the numerator (top number) and the denominator (bottom number) by each
prime number. The rules of divisibility will simplify this process:
Start with 2: EVEN numbers (ones that end with 2, 4, 6, 8, or 0) can be divided by two without a remainder (i.e., they
are divisible by 2).
Then go to 3: Find the SUM OF THE DIGITS (Add the digits together). If the sum can be divided by three then the
number is divisible by 3. [NOTE: Since you can tell by looking if a number is divisible by two or by five, you
may want to use the "eyeball approach" before checking three....]
Next try 5: Numbers that end with 5 or 0 are divisible by by five.
Go on to 7, 11, 13, 17 and so on. Unfortunately there is no easy way to determine whether the number will be
divisible by these -- you just have to try dividing by each. But, you can stop trying when the answer is less than
the divisor.
The most challenging fractions to simplify are generally the ones that don't look like they can be simplified. For
example:
Twenty-six can be divided by two (because it's even), but 65 can't. Sixty-five can be divided by five, but 26 can't... the
fraction looks like it can't be simplified --BUT WAIT-- factor either number and then try dividing by the other,
less obvious, factor. And always double-check to see if you can simplify ONE MORE TIME.
At the Paragraph Level
37. Patterns of Text: Functions of Language
Michael Halliday’s Functions of
Language (1975)
7 Language Functions
1. Instrumental
2. Regulatory
3. Interactional
4. Personal
5. Imaginative
6. Heuristic (investigate)
7. Representational
›Translate into these classroom
interaction patterns:
–Express an opinion
–Summarize
–Persuade
–Question
–Inform
–Sequence
–Disagree, agree, reach consensus
–Debate
–Evaluate
–Justify
38. Function Examples Classroom Experiences
Instrumental- language is used to
communicate preferences, choices,
wants, or needs
"I want to ..." Problem solving, gathering materials,
role playing, persuading
Personal- language is used to
express individuality
"Here I am ...." Making feelings public and
interacting with others
Interactional- language is used to
interact and plan, develop, or
maintain a play or group activity or
social relationship
"You and me ...."
"I'll be the cashier, ...."
Structured play, dialogues and
discussions, talking in groups
Regulatory- language is used to
control
"Do as I tell you ...."
"You need ...."
making rules in games, giving
instructions, teaching
Representational- language used to
explain
"I'll tell you."
"I know."
Conveying messages, telling about the
real world, expressing a proposition
Heuristic- language is used to find
things out, discover, wonder, or
hypothesize
"Tell me why ...."
"Why did you do that?"
"What for?"
Question and answer, routines,
inquiry and research
Imaginative- language is used to
create, explore, and entertain
"Let's pretend ...."
"I went to my grandma's
last night."
Stories and dramatizations, rhymes,
poems, and riddles, nonsense and word
playwww.communityinclusion.org/elm/Professionals/.../Halliday-handout.docx
39. Functions Common in Math Texts
› Sequential language
› Process-oriented
› Comparatives
› Argument
› Cause & effect
› Narrative (word problems)]
› Commands and directives http://2.bp.blogspot.com/-T93pPfGa0jM/UhrQ-
vsZtUI/AAAAAAAAHhU/wYHNglnd0qQ/s1600/cause+and+effect+Linguisti
c+patterns.jpg
40. Task: Find a Function of Language
›Directions:
1. Find a function in your text.
2. Answer these questions:
– Which one is it?
– Does the text explain how to do
it?
– Does the text present a model
of the language needed?
41. How do you select math academic language?
1. Identify functions of language embedded in
or congruent with the math topic and task.
2. Evaluate the enabling skills necessary to
complete academic tasks and determine if
the students have the academic language
to complete them. If not, explicitly teach
students the language to perform the tasks.
3. Provide sentence starters and dialogues to
teach and model the academic language
function.
4. Provide interactional practice to discuss
math reasoning.
5. Allow students options in selecting
academic language to learn and practice.
6. Adapt content, simplify or elaborate, when
the language is too complex and not
essential. Be sure to focus on the big
picture.
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ak0.pinimg.com/736x/c3/ee/1a/c3ee1a56e5314ffb
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42. REFERENCES
› Beck, I., McKeown, M., & Kucan, L. (2013). Bringing words to life, 2nd ed. NY, NY: Guilford Press.
› Bernier, A. (1997). The challenge of language and history: Terminology from the student optic.” In M. A. Snow and D. M. Brinton (eds.) The Content-Based
Classroom : Perspectives on Integrating Language and Content (1st Ed.) (pp. 96-97). New York: Longman.
› Carlo, M. S., August, D., & Snow, C.E. (2005). Sustained vocabulary: Strategy instruction for English language learners. In E. F. Heibert and M. L. Kamil
(Eds.), Teaching and Learning Vocabulary: Bringing Research to Practice (pp. 173-154). Mahwah, NJ: Lawrence Erlbaum.
› Cazden, C. B. (2001). Classroom discourse: The language of teaching and learning. Portsmouth, NH: Heineman.
› Chamot, A. U. (2009). The CALLA handbook: Implementing the cognitive academic language learning approach (2nd ed). NY, NY: Addison-Wesley.
› Chapin, S. H., O’Connor, C. & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn (2nd Ed.). Sausalito, CA: Math
Solutions Publications.
› Day, R. R., Omura, C, Hiramatsu, M. (1991). Incidental EFL vocabulary learning and reading. Reading in a Foreign Language, 7(2): 541-551.
› Li, W. E. [李文瑤]. (2012). Genre analysis of word problems in junior secondary school mathematics textbooks for ESL learners in Hong Kong. (Thesis).
University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5541038
› Folse, K. (2004). Vocabulary Myths: Applying Second Language Research to Classroom Teaching. Ann Arbor: University of Michigan Press
› Herbel-Eisenmann, B., Steele, M., & Cirillo, M. (2013). Developing teacher discourse moves: A framework for professional development. Mathematics
Teacher Educator, 1(2), 181-196.
› Herbel-Eisenmann, B. A., & Otten, S. (2011). Mapping mathematics in classroom discourse. Journal for Research in Mathematics Education, 42, 451-484.
› Marzano, R. J. (2004). Building background knowledge for academic achievement: Research on what works in schools. Alexandria, VA: ASCD.
› Marzano, R., & Pickering, D. (2005). Building academic vocabulary: Teacher’s manual. Alexandria, VA: ASCD.
› McKeown, M., Beck, I., Onanson, R., & Pople, M. (1985). Some effects of the nature and frequency of vocabulary instruction on the knowledge and use of
words. Reading Research Quarterly, 20: 522-535.
› Moschkovich, J. (2008). I went by twos, he went by one: Multiple interpretations of inscriptions as resources for mathematical discussions. The Journal of the
Learning Sciences, 17(4), 551–587.
› Nathan, M.J., Long, S.D., & Alibali, M.W. (2002). The symbol precedence view of mathematical development: A corpus analysis of the rhetorical structure of
algebra textbooks. Discourse Processes, 33(1), 1-21.
› Nation, I.S.P., & Waring, R. (1997). Vocabulary size, text coverage, and word lists. In N. Schmitt and M. McCarthy (Eds.), Vocabulary: Description, Acquisition
and Pedagogy (pp. 6-19). Cambridge: Cambridge University Press.
› Scarcella, R. (2003). Academic English: A conceptual framework. Berkeley, CA: University of California Linguistic Minority Research Institute.
› Schmitt, N., & Carter, R. (2000). The lexical advantages of narrow reading for second language learners. TESOL Journal, 9(1), 4-9.
› Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23, 139-159.
› Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University
Press.
› Slavit, D., & Ernst-Slavit, G. (2007). Teaching mathematics and English to English language learners simultaneously. Middle School Journal (November): 4-
11.
› Wagner, D., & Herbel-Eisenmann, B. (2009). Re-mythologizing mathematics through attention to classroom positioning. Educational Studies in Mathematics,
72(1), 1-15.
43. Secondary Students:
Math for New Arrivals
*Brenda Custodio is a retired middle and
high school ESL teacher and school
administrator of a newcomer high school.
*She received her Ph. D. from Ohio State
University in TESOL and now works at the
university level preparing teachers for a
career in TESOL.
*She is the author of two books:
› Custodio, B. (2011).How to Design and Implement
a Newcomer Program. Boston: Allyn & Bacon.
- Custodio, B. & O’Loughlin, J. B. (2017) Students
with Interrupted Formal Education: Bridging
Where They Are and What They Need.
Thousand Oaks, CA: Corwin.
Brenda Custodio– ESL
Consultant, Ohio State
University,
custodio.1@osu.edu
44. Developing
Numeracy Skills
› Think of learning math as a
learning the language of numerary
› Start with a pretest to determine
current numeracy skills
› Offer development classes on
basic math skills and math
vocabulary
› Allow students to move to more
advanced math classes as their
skills increase
› Provide sheltered math for
students with strong math skills but
limited English ability
Keys Tips for Teaching
Numeracy Skills
45. Math is NOT Necessarily a
“Universal Language”
•Writing numerals varies with different languages
•Many countries reverse the use of the comma and the
decimal point ($2.300,25)
•Long division varies by country
•Some countries use different local or religious calendars
•Many countries use a 24-hour day
•Most countries put the day before the month
•Metric system used in every other country but US
46. Mathematics has its own Vocabulary
Multiple Meaning Words
› Table
› Line
› Plane
› Times
› Plot
› Angle
› Average
› Combine
› Figure
› Faces
Phrase Words for
Mathematical Relationships
› If
› Because
› Unless
› Alike, similar, same
› Different from
› Probably
› Not quite
› Always, all the time
› Never
47. Components of a Strong Numeracy Program
1.Math Vocabulary, Number Sense, and Math Symbols
2.Operations and Basic Math Skills
3.Fractions, Decimals, and Percents
4.Measurement
5.Data Analysis and Statistics
6.Other Math Concepts such as Geometry, Algebra, and
Word Problems
(These components require math vocabulary)
48. Challenges of Teaching Math to Newcomers
Cummins says that “mathematical concepts and operations
are embedded in language” and “must be taught explicitly if
students are to make strong academic progress in
mathematics.”
He also says that new English learners trying to learn a
new language at the same time as new math concepts
presents a “heavy cognitive load,” and without the strong
anchor of background knowledge, the task may be
overwhelming.
49. Seven Strategies for Numeracy Development
1.Help students look for patterns rather than solutions.
2.Stress that there are multiple ways to solve a problem.
3.Encourage peer-group collaboration.
4.Utilize visuals and manipulatives to teach abstract
concepts.
5.Ground new concepts in practical, real-life situations.
6.Encourage the use of journals or logs that require
students to explain their actions with written words.
7.Focus on math vocabulary and math sentences.
54. References
Ciancone, Tom. (1996) “Numeracy in the Adult ESL
Classroom. Toronto Board of Education.
Cummins, Jim. (2006) “Every Student Learns.” Scott
Foresman-Addison Wesley Mathematics Series, p. iv, xviii.
Custodio, Brenda (2011) How to Design and Implement A
Newcomer Program. Allyn and Bacon.
55. Challenges for
Grades 3-5 English
Learners: Sentence
and Discourse
Level in Word
Problems
› Former K-8 ESL/Special Education Adjunct Graduate
Professor
› NJ City University, Bilingual/Special Education
endorsement candidates
› Ohio State University, Bilingual Elementary Graduate
Candidates
She is the author of:
› O’Loughlin, J. B. (2010). Academic Language Accelerator.
New York, NY: Oxford University Press.
› O’Loughlin, J. B. (2013). “What Time is It?” in Gottlieb, M, &
Ernst-Slavit, G. Academic Language in Diverse Classrooms:
Promoting Content and Language Learning, Mathematics,
Grades 3-5. Thousand Oaks, CA: Corwin.
› Custodio, B. & O’Louglin, J. B. (2017). Students with
Interrupted Formal Education: Bridging Where They Are and
What They Need. Thousand Oaks, CA: Corwin.
Judith O'Loughlin–
Consultant, Language
Matters, LLC,
joeslteach@aol.com
56. Word Problems in Mathematics
• Present significant comprehension difficulties
• particularly challenging for English Language Learners
(ELLs)
• These comprehension challenges have to do with
• lexical complexity
• sentence complexity
• Permeate mathematics textbooks and
standardized tests
• Seen as important measures of mathematical
understanding
57. Word Problems - Discourse Level - Addition
›Basic Elementary Word Problems - Addition
› Someone had ____ and ____. How much did she have:
– in all,
– together,
– combined,
– in total?
› Example: Sarah earned 58 dollars last week from her paper route.
This week she earned 47 dollars. How much money did she earn
for both weeks combined?
58. Word Problems - Discourse Level - Addition
›Elementary Word Problems - Addition
›Adding to an existing amount is challenging.
›Someone has ____ and bought _____ and ____.
›Example: While building the house, Charlie noticed that they were
running out of nails, so he told his father he’s going to buy some. If
they still have 9 nails left and Charlie bought 2 boxes of nails, one with
55 nails, the other with 31 nails, how many do they have now?
59. Word Problems - Discourse Level - Subtraction
›Basic Elementary Word Problems: Subtraction
›Linear Structure - Whole to Parts
›There are ______ and ______ are _____ . How many are not?
›Looking from the whole to the part.
›Example: Jan has 20 apples and 7 are red. How many are not red?
60. Word Problems
Mixed Addition and Subtraction Word Problems
›Two step problems
›Total amount needed is _______. First person does/makes ________
and a second person does/makes _____. How many should the third
person do/make?
›Example: A catering service needs to prepare 500 pieces of fish fillets
for the mayor’s 50th birthday party. Team one prepares 189 pieces
and team two prepares131 pieces. How many pieces does team three
have to prepare?
61. Word Problems
Estimating - Addition
›Story problem provides specific numbers. However, students are
asked to estimate the total.
›First person has______ and second person has _____. Estimate how
many all together.
›Example: Thirteen students from Mrs. D’s class want to go camping.
Eighteen students from Mrs. C’s class want to go camping. Estimate
how many students want to go camping altogether.
62. Word Problems
Multiplication
›Story problems with cost (money) and number of items.
›Utilize groupings/categories (containers, teams, classes, rows, rolls,
foods, etc.) and number of items in each.
›There are ___ (name of container) and _____ in each. How many in
all?
›Example: There are seven girls on stage. Each girl is holding nine
flowers. How many flowers are there in all?
63. Word Problems
Multiplication - Wording Issues
› One day Michelle decided to count her savings. She opened her
piggy bank and sorted out different coins and dollar bills. If she
counted a total of 20 nickels (a nickel is equivalent to 5 cents) what
is the total value of money does she have in nickels?
› She also found that she has 10 pieces of $5 bills. What is the total
value of money does she have in $5 bills?
64. Word Problems
Division
›Take the whole and determine number of parts.
›There were ____ (total number of items) then determine
how many items would each person/object contain.
›Examples: Eighteen fish were caught on a deep-sea fishing
boat. If each person on the boat caught 2 fish, how many
people were on the boat?
›A total of 8 people paid a total of $24 for admission. If each
ticket cost the same amount, how much did each ticket cost?
65. Word Problems
Mixed Operations
›Addition and Division:
–Greg has 91 erasers and Jane gives him 8 more. Greg gives each
of his 9 friends an equal number of erasers. How many erasers
does each friend get?
›Addition and Subtraction:
–Jane bought a large cheese pizza. The pizza was divided into 12
slices. Jane ate 2 slices, Marvin at 3 slices, and Mary ate one slice.
How many slices did they eat together? How many slices were left
over?
66. Word Problems
Elapsed Time
›Elapsed time problems have a starting time when an action or event
started and an ending time.
›Simple calculations, such as subtraction, will not provide an accurate
answer.
›Examples:
–Alexa went to the bookstore at 5:45 PM. She left the bookstore at
9:10 PM. How long was Alexa at the bookstore?
–Henry leaves arrives at work at 8:15 AM. He eats lunch at 12:00 PM.
Then returns to his desk at 1 PM and works until 4:30 PM. How long
does Henry work in the morning? In the afternoon after lunch?
67. Word Problems
Elapsed Time
Solving Elapsed Time Problems: Tools
› Manipulatives: Student clocks
› Story Board Frames to Sequence Elapsed Time
› 5:45 9:10 PM
69. Literacy Strategies Help ELs Learn Math
›Personal Math Dictionaries- key vocabulary words in math
problems.
›Math Diaries- Create your own math stories.
›Strategy Flash Cards or Book Marks- Create cards with simple
directions and examples
–Crossing Out Distractors
–Highlighting Key Operation Words and Numbers
–Think Aloud Prompts and Sentence Starters
›Comic Book Templates- Draw the Problem
›Manipulatives- counters, number cubes, dice, tangrams, rods, unifix
cubes, etc.
70. Bedtime Math App (in the app store)
www.BedtimeMath.org (online)
71. Picture Books:
Building Background for Math Instruction
› Pastry School in Paris, C. Neuschwandeer
› Perimeter, Area, and Volume, David A. Adler
› Full House: An Invitation to Fractins, D.A. Dodds
› A Very Improbable Story, E. A. Einhorn
› Sir Cumference (geometry picture book series), C. Neushwandeer
› Equal Shmequal: A Math Adventure, V. Kroll
› How Many Seeds in a Pumpkin? M. McNamara
72. Picture Books:
Building Background for Math Instruction
› Twizzlers Percentages, J. Pilotta and R. Bolster
› The Grapes of Math, G. Tang and H. Bridges
› Multiplying Menace: The Revenge of Rumplestiltskin, P. Calvert and
W. Geehan
› The Wishing Club: A Story About Fractions, D. J. Napoli and A. Curry
› Equal Schmequal, V. Keroll and P. O’Neill
› A Place for Zero, A. Sparagna LoPresti and P. Hornung
› Perimeter, Area, and Volume, D. A. Adler and E. Miller
73. References
›O’Loughlin, J. B. (2013). Grade 3: What Time is It? in (Gottlieb, M. & Ernst-Slavit, G. (Eds.),
Academic Language in Diverse Classrooms: Promoting Content and Language Learning.
Thousand Oaks, CA: Corwin.
› Word Problem Examples:
– Reading and Math Worksheets. K-5
– Learning:www.k5learning.com
› Spectrum Math Series. (2015) Greensboro, NC: Carson-Dellosa Publishing Group.
› Apps for Math Problem Practice
– Bedtime Math (www.BedtimeMath.org)
– Interactive Telling Time
– Khan Academy
– Pizza Fractions (Ages 6-8)
– Splash Math: Third Grade math, Mutlitplication, Fractions, and More
– Telling Time - Little Matchups Game
74. A Framework for
Analyzing Word
Problems: Beyond
Key Words
Chair and Associate Professor, Dept of Teaching and
Learning, University of Miami
Incoming President-Elect, TESOL International
Association
Publications – math-related
de Oliveira, L. C., Sembiante, S., & Ramirez, J. A. (in press). Bilingual academic
language development in mathematics for emergent to advanced bilingual students.
In S. Crespo & S. Celedón-Pattichis, & M. Civil (Eds), Access and equity: Promoting
high quality mathematics in grades 3-5. National Council of Teachers of Mathematics
(NCTM).
de Oliveira, L. C. (Series Ed.) (2014/2016). The Common Core State Standards and
English Language Learners. Alexandria, VA: TESOL Press.
- Civil, M., & Turner, E. (Eds.) (2014). The Common Core State Standards in
Mathematics for English Language Learners: Grades K–8.
- Bright, A., Hansen-Thomas, H., & de Oliveira, L. C. (Eds.) (2015). The Common
Core State Standards in Mathematics and English Language Learners: High
School.
de Oliveira, L. C. (2012). The language demands of word problems for English language
learners. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching:
Advancing mathematics education for ELLs (pp. 195-205). Reston, VA: National
Council of Teachers of Mathematics.
de Oliveira, L. C. (2011c). In their shoes: Teachers feel like English language learners
through a math simulation. Multicultural Education, 19(1), 59-62.
de Oliveira, L. C., & Cheng, D. (2011). Language and the multisemiotic nature of
mathematics. The Reading Matrix, 11(3), 255-268.
Kenney, R., & de Oliveira, L. C. (2015). Building functions from context: A framework for
connecting ELLs’ understandings of natural language and symbol sense in algebra. In
A. Bright, H. Hansen-Thomas, & L. C. de Oliveira (Eds). The Common Core State
Standards in Mathematics and English Language Learners: High School (pp. 57-70).
Alexandria, VA: TESOL Press.
Kenney, R., & de Oliveira, L. C. (2015). The role of symbol sense in mathematical
semiotic systems for English language learners. Teaching for Excellence and Equity
in Mathematics, 6(1), 7-15.
Luciana de Oliveira–
Chair and Associate
Professor, University of
Miami,
ludeoliveira@miami.edu
75. Presentation
›Demonstrate how teachers can
–identify these challenges
–focus on developing academic literacy and
mathematics content simultaneously
›Based on de Oliveira (2012) and de Oliveira
(2016) – a language-based approach to content
instruction (LACI)
76. Framework for Analyzing Word Problems at
the Elementary Level
›5 Qs to help teachers
–identify the language demands of word problems
before they present it to students
–get a better sense of the structure of the word problem
and the language demands for ELLs
›simultaneous focus on
–the mathematical concepts integrated in the word
problem
–aspects of language with which ELLs may have
difficulty
77. Framework for Analyzing Word
Problems at the Elementary Level
Guiding Qs to Ask Language Demands
to Identify
Tasks for Teachers
to Perform
1. What task is the
student asked to
perform?
Type of questions and
their structure – e.g.
how many, how much
To analyze the question
by identifying what it is
asking
2. What relevant
information is
presented in the word
problem?
Overall clause
construction – the verbs
and who, what, to
whom
To break down the
clause by finding what
information is
presented
3. Which mathematical
concepts are presented
in the information?
Specific clause
construction –
numerical information
presented in different
parts of the clause
To connect the
mathematical concepts
needed by looking for
specific numerical
information presented
in the clause
78. Guiding Qs to Ask Language Demands to
Identify
Tasks for Teachers to
Perform
4. What mathematical
representations and
procedures can students
use to solve the problem
based on the information
presented and the
mathematical concepts
identified?
Question + overall clause
structure + specific clause
structure
To connect all previously
analyzed pieces to
determine a variety of
mathematical
representations and
procedures that can be
used to solve the problem
5. What additional
language demands exist
in this problem?
Language “chunks”: nouns,
verbs, prepositional
phrases within clauses -
not as isolated elements
Connections between
clauses to determine how
different parts of the word
problem are connected.
To identify any aspect of
language that seems
problematic for ELLs not
recognized through the
previous guiding
questions
79. Word Problems and Key Language
Challenges
Word Problem 1:
• Indiana Statewide Testing for Educational Progress-Plus
(ISTEP+) Item Sampler
• 3rd grade (Indiana Department of Education, 2002)
• items = examples of the types of problems typically found
• academic standards for Grade 2 assessed in Grade 3:
• number sense, computation, algebra and functions,
geometry, measurement, and problem solving
• Example: sample item for the problem solving section
80. Word Problem 1
Denise is buying candy for 3 of her friends. She wants to give each
friend 4 pieces of candy. If each piece of candy costs 5¢, how much
money will Denise spend on candy for her friends?
Clause 1: Denise is buying candy for 3 of her friends.
Clause 2: She wants to give each friend 4 pieces of candy.
Clause 3: If each piece of candy costs 5¢,
Clause 4: how much money will Denise spend on candy for her
friends?
81. How teachers can analyze the language demands of
the word problem by focusing on each guiding
question:
Guiding Questions Language Demands of the Word Problem
1. What task is the
student asked to
perform?
• Look at the WP, find the question mark (?), look
closely at what the Q asks students to do
• The -if clause “if each piece of candy costs 5¢”
• The symbol ¢
• The construction spend on candy (vs. the book is
on the table)
• Human participant:
Denise→ the one who will be spending money;
“for her friends” →indicate Denise’s 3 friends
2. What relevant
information is
presented in the
word problem?
• Different important aspects of the text:
what is being presented in terms of the
problem (see the following table)
82. Clauses and Relevant Information Provided in
Word Problem 1
Clause Relevant Information Provided
Denise is buying candy
for 3 of her friends.
Who? = Denise
What is she doing? = is buying
What? = candy
For whom? = 3 of her friends
She wants to give each
friend 4 pieces of
candy.
Who? = She [Denise]
What does she want? = wants to give
To whom? = each friend
What? = 4 pieces of candy
If each piece of candy
costs 5¢
What? = each piece of candy
How much is each piece of candy? = 5¢
83. Guiding Questions Language Demands of the Word Problem
3. Which
mathematical
concepts are
presented in the
information?
• For Word Problem 1, our analysis of the question
reveals that it is asking about a quantity (how
much money)
• The following table helps teachers connect the
information and the mathematical concepts
presented in the problem (based on Huang &
Normandia, 2008)
Information Provided Mathematizing the Problem
Situation
Clause 1: Denise is buying candy for 3
of her friends.
Total number of friends = 3
Clause 2: She wants to give each
friend 4 pieces of candy.
Number of pieces of candy = 4
Each friend = 1
Therefore, 4 pieces of candy for 1
friend
Clause 3: If each piece of candy costs
5¢
Price for each (1) piece of candy = 5¢
84. How teachers can analyze the language demands of
the word problem by focusing on each guiding
question:
Guiding Questions Language Demands of the Word Problem
4. What mathematical
representations and
procedures can be
used to solve the
problem based on the
information
presented and the
mathematical
concepts identified?
• Connect the information, the mathematical concepts
presented in the problem, and the mathematical
representations and procedures that can be used to
solve the problem
• The following table helps teachers draw on the
crucial mathematical language and ideas inherent in
the construction of the word problem to plan lessons
that enhance ELLs’ having access to the problem
85. Information Provided Mathematical
Concepts
Mathematical
Representations and
Procedures
Clause 1: Denise is buying
candy for 3 of her friends.
Total number of friends =
3
Addition 4 + 4 + 4: 3 groups
of 4 because there are 3
friends.
4 + 4 + 4 = 12 pieces of candy
total that Denise will buy
Clause 1 Representation
Friend 1
Friend 2
Friend 3
Clause 2 Representation
Friend 1
Friend 2
Friend 3
Clause 2: She wants to
give each friend 4 pieces
of candy.
Number of pieces of
candy = 4
Each friend = 1
Therefore, 4 pieces of
candy for 1 friend
12 pieces of candy
1 piece of candy
86. Information Provided Mathematical
Concepts
Mathematical
Representations and
Procedures
Clause 3: If each piece of
candy costs 5¢
Price for each (1) piece of
candy = 5¢
1 piece of candy = 5¢
12 pieces of candy total
that Denise will buy
5 + 5 + 5 + 5 + 5 + 5 + 5
+5 + 5 + 5 + 5 + 5 = 60¢
Clause 3 Representation
Friend 1
Friend 2
Friend 3
5¢ 5¢ 5¢ 5¢
5¢ 5¢ 5¢ 5¢
5¢ 5¢ 5¢ 5¢
12
groups
of 5¢
87. How teachers can analyze the language demands of
the word problem by focusing on each guiding
question:
Guiding Questions Language Demands of the WP
5. What additional
language demands
exist in this problem?
• Identify language “chunks” such as a combination of
nouns, verbs, and prepositional phrases within
clauses → relationships of language chunks to other
parts of WP
• Reference devices “words that stand for other words
in a text” (Schleppegrell & de Oliveira, 2006, p. 263)
For example, the human participant Denise:
Denise in Clause 1
She in Clause 2
The noun group 3 of her friends in Clause 1
these participants are picked up in Clause 2 as
each friend
these participants are picked up in Clause 4 as
her friends
88. Language-based Approach to Content
Instruction LACI
› Refers to a simultaneous focus on language and
content
› Content-area instruction: major context for
building language and literacy in the education of
ELLs
(August & Shanahan, 2006)
–Need to fully understand the demands of mathematics
discourse for ELLs
–Need to clearly consider the language demands of
mathematics at the elementary school level
› discipline-specific academic support in language
and literacy development
89. LACI
›ELLs need to be able to engage with the
meanings presented
›Content is never separate from the language
through which that content is manifested
›Description of potential linguistic challenges and
framework can help teachers:
–be more proactive in helping ELLs learn the ways
language is used to construct mathematical knowledge
–better understand the language demands and address
them in their curriculum
“accessibility”
90. LACI
Accessibility
›making texts more accessible means more than
simplifying the language through which content is
manifested
›ELLs need access to mathematics discourse
›A matter of social justice
–If ELLs are not given opportunities to engage and
participate in experiences involving the use of appropriate
mathematics discourse, they will continue to be at a
disadvantage.
91. Why Use LACI in mathematics?
›Description of the language demands of WP
based on guiding questions and framework can
help teachers:
–analyze the construction of word problems
–make content accessible to ELLs by providing them
access to the ways in which knowledge is constructed
in mathematics
–not to simplify these word problems but to enhance
teachers’ understandings about how mathematical
disciplinary discourse is constructed in them
92. References
› de Oliveira, L. C. (under contract). A language-based approach
to content instruction (LACI) for English language learners:
Academic language in the content areas. University of Michigan
Press.
› de Oliveira, L. C. (2016). A language-based approach to content
instruction (LACI) for English language learners: Examples from
two elementary teachers. International Multilingual Research
Journal, 10(3), 217-231.
› de Oliveira, L. C. (2012). The language demands of word
problems for English language learners. In S. Celedón-Pattichis
& N. Ramirez (Eds.), Beyond good teaching: Advancing
mathematics education for ELLs (pp. 195-205). Reston, VA:
National Council of Teachers of Mathematics.
› de Oliveira, L. C., & Cheng, D. (2011). Language and the
multisemiotic nature of mathematics. The Reading Matrix, 11(3),
255-268.
93. Elementary School
›[Mathematics textbook]
›Three sisters attended a movie that cost $5 per person.
Each sister spent $2 on popcorn. Their mother gave them
$30 to spend for all three. How much money was left?
94. Middle School
›Middle School: [Section 3 Sequences and Equivalent Equations. This section explores
patterns in mathematics and in nature to help students find patterns and use them to make
predictions as a problem-solving strategy]
›At the end of February, Sarah began to save for a $340
mountain bike. At the time she had $173 in her savings
account. Her savings increased to $208 in late March and
$243 in late April. If her savings pattern continues, when
will she be able to buy her bike?
95. High School Level
›[Section 2-8 Explaining Multiplication and Division Related Facts. Chapter 2 is about using “algebra to
explain” by examining algebraic properties, such as distribution and commutativity. This section seems
to focus on using commutativity to find “facts.” This problem is the last one in the section before review
problems are started]
›Meli went grocery shopping. Her least expensive purchase
was a drink. She bought bread which cost twice as much
as the drink, salad that was four times as much as the
drink, and laundry detergent that was five times as much
as the drink. Her bill came to $18. How much did each item
cost Meli?
96. Thank you for your attention and participation!
› Feel free to contact us.
– Judith O'Loughlin– Consultant, Language Matters, LLC,
joeslteach@aol.com
– Kate Reynolds– Title III Grant Curriculum Designer, Missouri State
University, katerey523@gmail.com
– Brenda Custodio– ESL Consultant, Ohio State University,
custodio.1@osu.edu
– Luciana de Oliveira– Associate Professor, University of Miami,
ludeoliveira@miami.edu
